Power of a power, and zero and negative indices

Picture a map that shows ten times more ground with every notch you zoom out. Zoom out three notches and you see a thousand times more. Ten, times ten, times ten. Now take that whole three-notch jump and do it four times over. Three and four are the only numbers on the page. But you have not moved seven notches. You have moved twelve. Going the other way is stranger still. Zero notches leaves you exactly where you were, so the scale is one, not nothing. And a backwards notch never turns a number negative. It flips it into a fraction. Three rules, all hiding in one small index number.

Start with the map jump. A power, raised to another power.

First, a contrast. In the video on multiplying and dividing powers, the rule is add the indices. Two cubed times two to the power four is two to the power seven.

This is a different shape on the page. Two cubed, all raised to the power four. Two cubed is one block: three twos multiplied together. Raising that block to the power four means four copies of the whole block.

So count the twos. Three in every block, and four blocks. Three times four is twelve. Two to the power twelve.

Put those two side by side. Two cubed times two to the power four gives two to the power seven. Two cubed, all raised to the power four, gives two to the power twelve. Same two numbers, three and four. Add when you are multiplying powers. Multiply when a power is raised to a power.

Here is the line to keep. Copies of copies. Zero is one. Minus flips. The first part carries straight on from the video about multiplying and dividing powers. Count the copies, and the base holds still.

Your turn, and this one should feel easy. Three squared, all raised to the power three. Is it three to the power five, three to the power six, or nine to the power six?

Have a think. I'll wait.

The answer is three to the power six.

Two threes in every block, three blocks. Two times three is six. Three to the power five comes from adding instead of multiplying. Nine to the power six comes from squaring the base as well, and the base holds still. Three to the power six is seven hundred and twenty nine.

Now the strange one. An index of zero.

Take six to the power seven, divided by six to the power seven. The same number on the top and the bottom. Anything divided by itself is one. No rule needed for that, it is just true. Now run the same division through the index rule. In the video on multiplying and dividing powers, you subtract the indices. Seven take away seven is zero. So the answer is six to the power zero. Two ways of reading one division. So six to the power zero has to be one.

That holds for every base except zero itself. Anything to the power zero is one. Learn it as a plain fact. The division is there so you know it was not invented on a whim.

Quick one. Work out eight to the power zero.

Work that one out in your head. I'll wait.

The answer is one.

Not eight. Not zero. One. The base makes no difference here. Eight, a hundred, or a million. Power zero, answer one.

Last rule. A minus sign in the index.

Fix one thing before the example. A minus in the index never makes the answer negative. Never. It is not a minus sign attached to the number. It is an instruction to flip.

Back to base six. Six to the power seven, divided by six to the power nine. Seven sixes on the top. Nine sixes underneath. Cancel seven off the top against seven from the bottom. The top is used up, so it leaves a one. Two sixes are still sitting underneath. One over six squared. One over thirty six. Subtract the indices instead and you get seven take away nine, which is minus two. Six to the power minus two. Same division, so six to the power minus two is one over thirty six. A fraction. Positive, and small.

There is the rule. A minus index means take the reciprocal, which is just turning it upside down. Six to the power minus two is one over six squared.

Your turn on the harder one. Write four to the power minus two as a fraction. Is it minus sixteen, one over eight, or one over sixteen?

Pause it there and work it out. I'll wait.

The answer is one over sixteen.

Flip it first: one over four squared. Then finish the job. Four squared is sixteen, so it is one over sixteen. Minus sixteen is the trap, because the minus lives in the index, not on the answer. One over eight would be four times two, and four squared is four times four.

Now the exam side of these three rules.

Three shapes to recognise on the page. Brackets round a power with an index outside: multiply the two indices. An index of zero: write one. A minus index: flip it into a fraction.

Two habits that finish the answer off. If there are brackets, check that you are multiplying and not adding. And if the question says as a fraction, do the bottom line as well. One over four squared is not finished until you have written one over sixteen.

Two to try, without a calculator. Six to the power zero. Then two to the power minus three, written as a fraction.

Pause and work them out. I'll wait.

Six to the power zero is one. Two to the power minus three is one over eight.

Two cubed is eight, and the minus flips it. One over eight.

Three cases, one page. Here they are.

A power raised to a power: multiply the indices. Two cubed, all raised to the power four, is two to the power twelve, because you are counting copies of copies. An index of zero gives one, whatever the base. Six to the power zero is one. Eight to the power zero is one. A minus index flips the number into a fraction. Six to the power minus two is one over thirty six. Small, and positive. Never negative.

Copies of copies. Zero is one. Minus flips.

Next in the chapter: fractional indices, which is Higher tier only. If you are sitting Foundation, skip ahead to converting to and from standard form. You will not miss anything.

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